We study the optimal contract between well-diversified risk-neutral shareholders and a constant relative risk aversion (thereafter CRRA) manager in a continuous-time agency model. In our model, the manager controls the instantaneous growth rate of an output process. Cost of effort is assumed as a standard quadratic function. A number of interesting results are obtained.
First, the wealth effect exhibited by CRRA utility plays an important role in determining the optimal contract. We find that the optimal contract is strictly increasing but concave in the output value if the manager is more risk-averse than a log-utility manager. The reason is that the marginal utility from consumption is lower for a higher wealth level and more so for a more risk-averse utility. Relative to a less risk-averse manager (a log-utility manager), it is more effective to compensate a more risk-averse manager with a contract which has a relatively higher portion in equity-linked compensation at the lower end of the output value than at the higher end. However, a reduced equity-linked compensation reduces effort. In our model, effort affects the output scale rather than level as in, e.g., Holmstrom and Milgrom (1987). This multiplying effect of effort calls for higher effort level if the current value of the output process is higher.
But a higher current output value implies a likely higher final output value, which in turns implies a higher compensation, resulting in a lower marginal utility. Therefore, there is a trade-off between the benefit of reducing the equity proportion at high output value due to the wealth effect and the cost of reduced effort (in the form of lower output value). Indeed, the wealth effect of lower marginal utility and multiplying effect of effort offsetting each, yielding an optimal linear contract when (1) the manager has a log-utility; and (2) there is no lower bound on the compensation. For a more risk-averse manager, the wealth effect dominates and a concave compensation becomes optimal. The concavity of the optimal compensation is in sharp contrast with the convexity of compensations such as stock options whose optimality is normally not established.
If, on the other hand, the managerial compensation has a positive lower bound (which can be interpreted as a base payment or limited liability), the optimal compensation consists of a cash payment plus an equity-linked component which has a nonzero (positive) value only if the (terminal) output value is above a critical level. However, while the equity linked component has the appearance of a stock option (kinked payoff due to the lower bound constraint), it is generally not. In our model stock options become optimal under two conditions: (i) the manager has a log-utility; and (ii) his compensation has a positive lower bound.
The result that the constrained optimal compensation with a lower bound is option-like appears to be general and has also been obtained in, among others, Kadan and Swinkels (2008), and Jewitt, Kadan and Swinkels (2008). Even though these papers have all considered bounds on compensations, they differ in focuses. For example, while studying the effects of compensation bounds in static agency models is the focus of Jewitt, Kadan and Swinkels (2008), our paper, for the most part, is on the dynamics of optimal efforts and pay-performance sensitivity in a dynamic model without bound constraints. On the other and, while the focus of Kadan and Swinkels (2008) is on the effort incentives when compensation is in stocks or stock options, one of the objectives of our paper is to provide the conditions (provided in the previous two paragraphs) when compensation in stocks or stock options is optimal.
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Optimal Compensation and Pay-Performance Sensitivity in a Continuous-Time Principal-Agent Model
